“Procedural” values for cooperative games

Abstract

This paper introduces a new notion of a "procedural" value for cooperative TU games. A procedural value is determined by an underlying procedure of sharing marginal contributions to coalitions formed by players joining in random order. We consider procedures under which players can only share their marginal contributions with their predecessors in the ordering, and study the set of all resulting values. The most prominent procedural value is, of course, the Shapley value obtaining under the simplest procedure of every player just retaining his entire marginal contribution. But different sharing rules lead to other interesting values, including the "egalitarian solution" and the Nowak and Radzik "solidarity value". All procedural values are efficient, symmetric and linear. Moreover, it is shown that these properties together with two very natural monotonicity postulates characterize the class of procedural values. Some possible modifications and generalizations are also discussed. In particular, it is shown that dropping one of monotonicity axioms is equivalent to allowing for sharing marginal contributions with both predecessors and successors in the ordering.

Measuring power and satisfaction in societies with opinion leaders: an axiomatization

“…In the context of cooperative TU-games, Lehrer (1988) and Haller (1994) introduced properties that consider collusion of players. In particular, Haller (1994) considers different types of collusion neutrality properties requiring that the sum of the payoffs of two colluding players does not change, see also Malawski (2004). In van den Brink (2010) these properties are stated in terms of games in which the players belong to some hierarchical structure, the so-called games with a permission structure.…”

Reconciling marginalism with egalitarianism: consistency, monotonicity, and implementation of egalitarian Shapley values

“…Malawski (2004) obtains the egalitarian Shapley values by a procedure where for every order of entrance to the 'grand coalition' player i ∈ N gets a share α in its marginal contribution and the predecessors of i equally share the remainder of i's marginal contribution. 13 Taking the average over all orders of entrance yields the corresponding α-egalitarian Shapley value as expected payoffs.…”

Modification of Shapley Value and its Implementation in Decision Making

“…In his papers on simple games with empty cores, which refer to modeling of political and committee decision making, Shapley (1962aShapley ( , 1963Shapley ( , 1964a It is worth to mention a recent extension of the Shapley value to the so-called procedural value for transferable utility games proposed by Malawski (2013), where the players can share their marginal contributions with their predecessors. A somewhat similar solutions concept, called "egalitarian" value was recently investigated by Casajus and Huettner (2013).…”